---
kind: concept
slug: finite-element-method
title: Finite element method
summary: >-
  Discretises a continuous body into elements and solves its equilibrium or transient equations numerically. Variants: full 3D continuum elements, geometrically exact elements for rods (GE-FEM), and FE as reference data for learned surrogates. Trade-offs: the most physically complete option, but costly and too heavy for real-time control.
tags: [soft-robotics, robot-modeling, finite-elements, continuum-mechanics, surrogate-modeling]
papers: ["2112.03645", "2507.05291"]
related: [cosserat-rod-theory, graph-neural-network, physics-informed-neural-network, surrogate-modeling]
status: draft
updated_at: 2026-10-19
---

## Overview

Soft robots are continuous bodies, so their state is a field over the body
rather than a finite set of joint angles [arxiv:2112.03645 §II]. The finite
element method (FEM) approximates that field with a mesh of elements. For 3D
bodies the static equilibrium is written as Q_int(q) = Q_ext(q), where q are
the nodal unknowns, and transient problems add a dynamic ODE system
[arxiv:2112.03645 §IV.A]. Nonlinearity can come from the geometry (large
displacements) or from the material [arxiv:2112.03645 §IV.A].

The shared core is the discretisation of the body and the balance between
internal and external forces. Variants differ in the kind of body they
discretise and in how they handle finite rotations.

FEM is also used outside robotics, as the reference solver for nonlinear
continuum problems. In two-scale homogenisation (FE²) each macro integration
point needs its own micro solve, which is expensive [arxiv:2507.05291 §1].
A learned model can replace those micro solves, but FE stays the reference
against which it is measured [arxiv:2507.05291 §4.3] [arxiv:2507.05291 §5.3].

## Variants

### 3D continuum finite elements

The body is meshed in full three dimensions. The review lists this family
under continuum mechanics and says it suits the design of new concepts because
it covers a broad range of physics [arxiv:2112.03645 tab. 1]. Its main cost is
that the model has many degrees of freedom, which the review judges too heavy
for control [arxiv:2112.03645 tab. 1] [arxiv:2112.03645 §IV.A]. Card:
[Soft Robots Modeling](/papers/2112.03645).

Two formulations are named for capturing finite rotations: corotational and
ANCF (absolute nodal coordinate formulation) [arxiv:2112.03645 §IV.A]. The
review does not compare them.

For actuation, the review describes the SOFA handling of tendon-like actuators
as a linearised constraint, with Lagrange multipliers λ and a three-step
projection procedure [arxiv:2112.03645 §IV.A].

### Geometrically exact finite elements for rods (GE-FEM)

For slender, beam-like bodies the review places a geometrically exact finite
element formulation in the directors-based family. It is an energetic Ritz
method: it reduces the rod configuration to Lagrange ODEs [arxiv:2112.03645
§V.B]. The review names it as the main example of this reduction, next to
modal Ritz and GVS [arxiv:2112.03645 §V.B]. It is treated on the
[Cosserat rod theory](/concepts/cosserat-rod-theory) page, which covers the
rod model it discretises.

### FE as reference and data source for a learned surrogate

Here FE is not the model that runs in the loop. It produces the reference
stresses: the paper's targets are nodal stresses averaged from FE solutions of
periodic cells with a central hole [arxiv:2507.05291 §4.1] [arxiv:2507.05291
§2.3]. A graph neural network is then trained to predict the stress at every
node from the cell's mean stress [arxiv:2507.05291 §3]. The GNN is a learned
surrogate, not a finite element solver; its architecture is covered on the
[graph neural network](/concepts/graph-neural-network) and
[physics-informed neural network](/concepts/physics-informed-neural-network)
pages.

The FE mesh also gives the discrete stress divergence used as a training
penalty. Its operator is built from shape-function derivatives of the FE mesh
[arxiv:2507.05291 §2.3] [arxiv:2507.05291 eq. 31–35]. The same operator can
be used to check any stress field for local equilibrium, which is the
residual the paper adds as a training penalty [arxiv:2507.05291 §3.3].

Reported gains are author-reported and not like-for-like. On hyperelastic
meshes the surrogate's inference is about 500 times faster than FE, but the
timing compares GPU inference with CPU FE, excludes training, and uses no
batching [arxiv:2507.05291 §5.3.1]. On linear meshes it is faster at every
tested size, with no factor given [arxiv:2507.05291 §4.3.1]. Card:
[Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering
Finite Strain Hyperelasticity](/papers/2507.05291).

## When to use which

The review judges its options qualitatively. It gives no measured accuracy or
run-time figures, so the rows about the review cite its statements, not
benchmark results. The last row draws on the second card, whose figures are
author-reported.

| Situation | Use | Why |
|---|---|---|
| New soft-robot concept, full physics wanted, no real-time requirement | 3D continuum finite elements | The review lists design of new concepts as its typical use [arxiv:2112.03645 tab. 1] [arxiv:2112.03645 §IV.A]. |
| Slender, beam-like body that must run in real time or under control | GE-FEM (Ritz reduction of a rod) | The review lists beam-like robots, real-time simulation and control as typical uses of the directors family [arxiv:2112.03645 tab. 1] [arxiv:2112.03645 §V.B]. |
| Body is three-dimensional and must be simulated for control | Not a 3D FEM model directly | The review says full 3D FEM has too many degrees of freedom for control [arxiv:2112.03645 tab. 1]. A reduced model is needed [arxiv:2112.03645 §V]. |
| Many solves of a periodic micro-structure, and local (per-node) stress is needed, not only the average | FE for the reference data; a learned surrogate for the repeated solves, with an equilibrium penalty | The surrogate keeps per-node stress that average-field reduced models lose [arxiv:2507.05291 §1]. It needs a generated FE training set [arxiv:2507.05291 §4.1], and its speed-up excludes training [arxiv:2507.05291 §5.3.1]. |

## Implementation sketch

The review gives no algorithm and does not name the nonlinear solver to use.
The sketch below is our paraphrase of the static case in [arxiv:2112.03645
§IV.A]. The choice of solver is left open here.

```pseudocode
# Static equilibrium of a discretised body [arxiv:2112.03645 §IV.A]
# Input:  mesh of elements, initial nodal unknowns q0, external loads Q_ext
# Output: equilibrium nodal unknowns q with Q_int(q) = Q_ext

def static_equilibrium(mesh, q0, Q_ext):
    q = q0
    # residual: internal minus external generalised forces
    while not converged(q):
        # internal forces from the element strain energy (geometric or material)
        r = Q_int(mesh, q) - Q_ext
        # update q with a solver of your choice (not stated in the review)
        q = update(q, r)
    return q
```

## Contradictions

- **Discrete divergence of the FE reference.** The linear-case divergence
  values differ between the paper's text and its table. Section 4.3 gives
  4.56e-4 for the learned model and 9.43e-4 for FEM [arxiv:2507.05291 §4.3].
  Table 1 gives 3.23e-4 and 9.18e-4 [arxiv:2507.05291 tab. 1]. The paper does
  not explain the difference.
- **Learned model below the FE reference on divergence.** Both tables report a
  lower mean discrete divergence for the learned model than for FE: 3.23e-4
  against 9.18e-4 in the linear case [arxiv:2507.05291 tab. 1], and 1.56e-3
  against 1.69e-3 in the hyperelastic case [arxiv:2507.05291 tab. 2]. The paper
  gives no explanation, so read this as a reported number, not as evidence
  that the learned stress is closer to equilibrium than FE.
- The review does not compare the corotational and ANCF formulations, nor does
  it report numbers for the cost of 3D FEM against the other families
  [arxiv:2112.03645 §IV.A] [arxiv:2112.03645 tab. 1].

## Papers

- [Soft Robots Modeling: a Structured Overview](/papers/2112.03645): a review that sorts soft-robot models into four families and describes the finite element formulations for 3D continuum bodies and for rods (GE-FEM).
- [Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity](/papers/2507.05291): a graph neural network that predicts per-node stress of a periodic micro-structure from its mean stress, trained against FE data with a discrete-divergence penalty.